LISA should detect gravitational waves from tens to hundreds of systems containing black holes with mass in the range from 10 thousand to 10 million solar masses. Black holes in this mass range are not well constrained by current electromagnetic observations, so LISA could significantly enhance our understanding of the astrophysics of such systems. In this paper, we describe a framework for combining LISA observations to make statements about massive black hole populations. We summarise the constraints that LISA observations of extreme-mass-ratio inspirals might be able to place on the mass function of black holes in the LISA range. We also describe how LISA observations can be used to choose between different models for the hierarchical growth of structure in the early Universe. We consider four models that differ in their prescription for the initial mass distribution of black hole seeds, and in the efficiency of accretion onto the black holes. We show that with as little as 3 months of LISA data we can clearly distinguish between these models, even under relatively pessimistic assumptions about the performance of the detector and our knowledge of the gravitational waveforms.
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